PropositionStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Nonempty connected orientable manifolds have exactly two orientations

Statement

A nonempty connected orientable manifold has exactly two orientations; on a disconnected manifold the choices are componentwise.

Facts & Assumptions

Given: A nonempty connected orientable smooth manifold $M$ and one chosen orientation $o$ on it.

[L1]

An orientation is a smooth pointwise choice of a determinant-line ray (Oriented smooth manifolds and oriented charts).

[L2]

Orientability asserts the existence, but not a preferred choice, of such an orientation (Orientable manifolds).

Proof

technique · direct

1.1

By [L1], at each point any other orientation $o^{'}$ is either $o$ or its opposite. Smoothness of both ray choices makes the relative sign locally constant.

givenL1L2

2.1

Connectedness makes that sign constant, so $o^{'} = o$ everywhere or $o^{'} = - o$ everywhere. Both choices exist and are distinct because $M$ is nonempty. On a disconnected manifold the same locally constant sign may be selected independently on each component.

givenstep 1.1∎

Depends on

Used by

An orientable manifold has a canonical orientation False statement

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)